A topological Maslov index for 3-graded Lie groups - ResearchGate

A topological Maslov index for 3-graded Lie groups 27
Proof. Clearly the map ηE: C → V, x + iy ↦→ xe + yσ is a morphism
of involutive unital Jordan algebras, where the involution on C is complex
conjugation.
We recall from Theorem A.8 that gE is a Lie subalgebra of g. Since
gE is generated by E and τ(E), its center zE coincides with the centralizer
zE of E + τ(E), and the quotient g ′ E := gE/zE is an involutive 3-graded Lie
algebra whose 0-component has a faithful representation on E. From that it
easily follows that g ′ E
is isomorphic to the Tits-Kantor-Koecher Lie algebra
TKK(E) = TKK(C) ∼ = sl2(C) of the unital Jordan algebra C because it is
an A1-graded Lie algebra (cf. [Ne03, Ex. I.9(a),(c) for more details). Since all
central extensions of the simple Lie algebra sl2(C) are trivial, we conclude that
zE ∩ [gE, gE] = {0}, so that gE ∩ g0 = [E, τ(E)] implies zE = {0} and therefore
gE ∼ = sl2(C).
In Definition I.9 we have seen that the Lie algebra ge = span{e, τ(e), [e, τ(e)]}
with 1-dimensional grading spaces is isomorphic to sl2(R) with the involution
τe
a b −a c
= .
c d b −d
Since the grading spaces gE ∩gj are complex one-dimensional, it follows that ge
is a real form of the complex Lie algebra gE .
Next we determine the involution τE on gE ∼ = sl2(C) corresponding to the
restriction of τ to gE . Since the centroid
Cent(gE) = {ϕ ∈ End(g): (∀x ∈ gE) [ϕ, adx] = 0}
is isomorphic to C as an associative algebra, the involution τ induces a field
isomorphism τ ′ on Cent(gE). The involution τE is complex linear if this
isomorphism is trivial and it is antilinear otherwise. We denote the scalar
multiplication with i on sl2(C) by i, which is considered as an element of
Cent(gE). Then σ = i.e leads to τ.σ = τ ′ (i)τ(e). From
−ie = −σ = Q(e)σ = 1
2 [[e, τ.σ], e] = −1
2 (ade)2τ.σ = − 1
2 (ade)2τ ′ (i)τ(e)
= −τ ′ (i) 1
2 (ade)2τ(e) = τ ′ (i)Q(e)e = τ ′ (i)e
we derive τ ′ (i) = −i and hence that τE is antilinear.
Therefore τE is determined by its restriction to the real form ge, and hence
is the involution on sl2(C) for which η g σ
a b −a c
↦→
c d b −d
is a morphism of involutive Lie algebras.
Lemma IV.3. If G satisfies (A3), then the homomorphism η G σ : SL2(C) → G
integrating ηg σ maps −1 to 1.